Non-modal Stability Analysis of Thermocapillary Liquid Layers
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摘要: 利用非模态稳定性方法研究了亚临界情况下的热毛细液层对初始扰动和外加激励的敏感性。通过瞬态增长函数和反馈函数分别反映流场对初始扰动和外加激励的放大。研究结果表明, 小Prandtl数（Pr）下的亚临界流动对初始扰动和外加激励均十分敏感，最大扰动放大与Reynolds 数（Re）的平方近似成正比。在大Pr下，只有回流的亚临界流动存在对外加激励的显著放大，其最大值分别随Re5和Pr5呈线性增长。随着外加激励频率的增大，最优扰动波数逐渐减小。流场和温度场表明输出的扰动速度和温度量级远大于输入的量级，并且与管道流动相比存在明显不同。Abstract: The sensitivity of subcritical thermocapillary liquid layers to initial disturbances and external excitations is investigated by the non-modal stability analysis. The amplifications of initial disturbances and external excitations are measured by the growth function and response function, respectively. Results show that at small Prandtl numbers (Pr), the subcritical flows are sensitive to both initial disturbances and external excitations. The maximum amplifications are approximately proportional to the square of Reynolds number (Re). At large Pr, large amplifications to external excitations are found in the return flow. The maximum response increases linearly with Re5 and Pr5. When the frequency increases, the total wave number of the optimal response decreases. The flow and temperature fields indicate that the magnitudes of output temperature and velocity are far larger than those of input, which is significantly different from the pipe flow.
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Key words:
- Thermocapillary /
- Non-modal stability /
- Disturbance amplification
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图 2 线性流Pr = 0.002, k = 4,
$ \phi = {90^ \circ } $ 不同Reynolds 数下瞬态增长函数G(t)随时间t的变化Figure 2. Variation of G(t) with time of the linear flow at Pr = 0.002, k = 4,
$ \phi = {90^ \circ } $ with various Reynolds numbers
图 3 线性流Pr=0.002, k=4,
$ \phi = {90^ \circ } $ 最大瞬态增长Gmax和最大扰动放大$ {\Re _{{\text{max}}}} $ 随Reynolds数的变化Figure 3. Maximum transient growth function Gmax and the maximum response function
$ {\Re _{{\text{max}}}} $ versus Reynolds numbers of the linear flow at Pr=0.002, k=4,$ \phi = {90^ \circ } $ 
图 4 回流Re=2, k=2.3,
$ \phi = {0^ \circ } $ 不同Prandtl 数下扰动放大$ \Re$ 随实频率$ \omega $ 的变化Figure 4. Response function
$ \Re $ versus the real frequency$ \omega $ of the return flow at Re=2, k=2.3,$ \phi = {0^ \circ } $ with various Prandtl numbers
图 5 回流k=2.3,
$ \phi = {0^ \circ } $ 最大扰动放大$ {\Re _{{\text{max}}}} $ 随Prandtl数和Reynolds数的变化Figure 5. Maximum response function
$ {\Re _{{\text{max}}}} $ versus the Prandtl numbers and Reynolds numbers at k=2.3,$ \phi = {0^ \circ } $ 
图 6 线性流Pr=0.002、Ma=3在不同实频率下
$ \alpha - \beta $ 平面内${\rm{lg}}\Re $ 的等值线Figure 6. Level lines of the logarithm of the response
$ {\rm{lg}}\Re $ in the$ \alpha - \beta $ plane for the linear flow at Pr = 0.002, Ma=3 with various real frequency
图 7 回流Pr=150，Ma=300在不同实频率下
$ \alpha - \beta $ 平面内$ {\rm{lg}}\Re $ 的等值线Figure 7. Level lines of the logarithm of the response
$ {\rm{lg}}\Re $ in the$ \alpha - \beta $ plane for the return flow at Pr =150, Ma=300 with various real frequency
图 8 线性流Pr=0.002, Ma=3, k=2,
$ \phi = {90^ \circ } $ 扰动放大对应的扰动场。(a) 输入速度场，(b) 输出速度场，(c) 输入温度场，(d) 输出温度场Figure 8. Perturbation fields corresponding to the response for the linear flow at Pr=0.002, Ma=3, k=2,
$ \phi = {90^ \circ } $ . (a) Input velocity field, (b) output velocity field, (c) input temperature field and (d) output temperature field
图 9 回流Pr=150, Ma=300, k=2.3,
$ \phi = {0^ \circ } $ 扰动放大对应的扰动场。(a) 输入速度场，(b) 输出速度场，(c) 输入温度场，(d) 输出温度场Figure 9. Perturbation fields corresponding to the response for the return flow at Pr=150, Ma=300, k=2.3,
$ \phi = {0^ \circ } $ . (a) Input velocity field, (b) output velocity field, (c) input temperature field and (d) output temperature field -
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